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Bubble on Real Estate: The Role of Altruism and FiscalPolicy
Lise Clain-Chamosset-Yvrard, Thomas Seegmuller
To cite this version:Lise Clain-Chamosset-Yvrard, Thomas Seegmuller. Bubble on Real Estate: The Role of Altruism andFiscal Policy. 2018. �halshs-01880937�
Working Papers / Documents de travail
WP 2018 - Nr 21
Bubble on Real Estate: The Role of Altruism and Fiscal Policy
Lise Clain-Chamosset-YvrardThomas Seegmuller
Bubble on real estate: The role of altruism and fiscal policy
Lise Clain-Chamosset-Yvrard∗and Thomas Seegmuller†‡
Abstract
In this paper, we are interested in the interplay between real estate bubble, aggregate capitalaccumulation and taxation in an overlapping generations economy with altruistic households.We consider a three-period overlapping generations model with three key elements: altruism,portfolio choice, and financial market imperfections. Households realise different investmentdecisions in terms of asset at different periods of life, face a binding borrowing constraint andleave bequests to their children. We show that altruism plays a key role on the existence ofa productive real estate bubble, i.e. a bubble in real estate raising physical capital stock andaggregate output. The key mechanism relies on the fact that a real estate bubble raises incomeof retired households. Because of higher bequests, there children are able to invest more inproductive capital. Introducing fiscal policy, we show that raising real estate taxation dampenscapital accumulation.
JEL classification: E22; E44; G11.
Keywords: Bubble; Altruism; Real estate; Credit; Overlapping generations.
1 Introduction
The question we address in this paper is how we can explain that a real estate bubble can enhanceproductive capital and therefore aggregate output, whereas we could have in mind that investmentsin productive capital and in real estate could rather be substitutable. This question is importantregarding the consequences of the last financial crisis and also for the design of the fiscal policy.On the one hand, many countries have experienced real estate bubbles before the last financialcrises and, at the same time, a rise in capital stocks and aggregate output. Some empirical studies
∗Univ. Lyon, Universite Lumiere Lyon 2, GATE UMR 5824, F-69130 Ecully, France. E-mail: [email protected]†Corresponding author. Aix-Marseille University, CNRS, EHESS, Centrale Marseille, AMSE. E-mail:
[email protected]‡We are grateful for useful suggestions from Aurelien Eyquem, Frederic Jouneau-Sion, Xavier Raurich and
Bertrand Wigniolle. We also thank participants to the Workshop “Macroeconomics, finance and history of eco-nomics” held on June 2017 at Ecully, to the conference SAET 2017 and to the workshop “Real and financialinterdependencies: New approaches with dynamic general equilibrium models” held in Paris School of Economics onJuly 2017. Any remaining errors are ours. This work has benefited from the financial support of the French NationalResearch Agency (ANR-15-CE33-0001-01).
1
highlight that private non-residential investment in the United States increased significantly duringthe formation of the real estate bubble in the 2000s and then dropped when it burst (IMF, 2015).1
On the other hand, we note that the fiscal policy is especially designed to guide household savings.The french 2018 Finance Act (Loi de finances 2018), which replaces the solidarity tax on wealth by atax on real estate wealth, is a perfect illustration. By concentrating taxes on real estate, consideredas unproductive, a government aims to divert savings allocated from unproductive investment toproductive one in order to promote economic growth. As wealth mainly consists in real estate fora large part of households, and most of them invest in real estate to leave an inheritance to theirchildren, this idea that taxing real estate will necessarily promote productive capital could not holdfor the whole economy.
The aim of this paper is to exhibit a mechanism through which a rational real estate bubbleis accompanied by a boom in physical capital stock and in aggregate output. Our explanation isbased on the altruistic behaviour of households. Our framework will be further used to study themacroeconomic consequences of real estate taxation.
We consider an overlapping generations model with three-period lived households. Householdsrealise different investment decisions in terms of asset at different periods of life. When young,households can borrow and invest in real estate by buying land or a house for their old age. Whenmiddle-aged, they can save in the forms of deposits and physical capital to transfer wealth to theirold age.
In our paper, there are three key elements. First, assets play different roles in our model. Inaddition to serve as a collateral, real estate allows young households to transfer resources to their oldage. Accordingly, real estate can be seen as a long-term investment. Households behave as investorswho buy a real estate asset for resale purpose only, and not for housing services. The real estate assetis intrinsically useless, meaning that its fundamental value is zero. This asset is a bubble when it hasa positive value. Deposits and physical capital are both short-term investments. Indeed, householdsinvest in these assets when middle-aged and receive returns when old. Nevertheless, deposits arethe counterpart of loans and capital is used in production. This segmentation of asset marketsreflects the idea that households first invest in real estate, kept during all the life-cycle and usedto transfer bequests to children. Once this investment is reimbursed, agents are concerned withfinancial investments, through deposits and productive capital.
Second, we take into account financial market imperfections through a binding borrowing con-straint faced by young households. This contrasts with most of the recent contributions in thisliterature, which focus on financial frictions at the entrepreneur level (among others, Farhi andTirole, 2012; Martin and Ventura, 2012; Miao et al., 2015; Hirano and Yanagawa, 2017). In addi-tion, some empirical studies support our assumption (see Campbell and Mankiw, 1989; Jappelli,1990; Crook and Hochguertel, 2005). In our paper, households must pledge their labour incomeand their house as collaterals and, borrow against a part of their labour income and a part of themarket value of their real estate asset.2 An increase in the market value of the real estate asset ora higher labour income help young households to relax their borrowing constraint. In this paper,labour income can be seen as a fundamental collateral and the real estate asset as a bubbly one.3
1In their papers, Kamihigashi (2008) and more recently Miao et al. (2017) provide some historical examplesrevealing that the bursting of a real estate bubble is often followed by a recession.
2On the one hand, empirical evidence confirms that real estate collateral plays a significant role during crisis(Mera and Renaud, 2000). Moreover, macroeconomics model dealing with financial frictions consider a real estatecollateral constraint along the lines of Kiyotaki and Moore (1997) and Iacoviello (2005). On the other hand, somestudies argue that the debt limit also depends on household income (Marcet and Singleton, 1991; Ludvigson, 1998).
3Martin and Ventura (2011, 2016) use a quite similar constraint except that the fundamental collateral corresponds
2
The first type of collateral would rather represent the french system, while the second one wouldrather describe what happens in US. In addition, it will allow us to discuss the respective role ofeach type of collateral on the existence of a real estate bubble associated to a higher level of capitaland output.
Third, households behave altruistically by leaving bequests to their children. Several empiricalstudies support the idea that households have a desire to leave bequests and, underline its impor-tance for explaining saving behaviour (Kotlikoff and Summers, 1981; Dynan et al., 2004). In ourmodel, the desire to leave bequests are rationalized by a family altruism, defined as the situation inwhich parents care about their children’s income. As we will see, altruism is crucial for our results.
Our main concern is to investigate and understand the existence of a productive real estatebubble, i.e. a real estate bubble promoting capital accumulation and, thus aggregate output. Weshow that a real estate bubble occurs in a low interest rate environment. In this case, there existsan excess of savings in the economy, which provides a room for alternative stores of value. As thereal estate bubble provides a return equal to one at the steady state, this return creates an incentivefor households to hold the bubble and borrow to invest in this bubble.
Comparing the bubbleless and bubbly steady states, we show that a productive real estate bubbleoccurs when parents are sufficiently altruistic and the effects of fundamental and bubbly collateralscoming from the borrowing constraint are not too high. This result is consistent with empiricalevidence showing that productive investment increases significantly during the formation of a bubble(Martin and Ventura, 2012). Altruism plays a key role on the occurrence of productive real estatebubbles. In an economy without altruism, middle-aged households have not a high enough incometo sustain a higher level of capital in presence of a real estate bubble. Moreover, a productive realestate bubble exists whatever the type of collateral (fundamental or bubbly collateral) required bylender on the credit market. Surprisingly, the two types of collateral play quite similar roles andwe cannot conclude that one type further promotes the emergence of real estate bubbles.
Finally, we also investigate the effect of fiscal policy on the stock of physical capital and, hence,on aggregate output. We show that an increase of a real estate tax has a negative effect onproductive investment. An increase of the tax rate reduces the incentive to buy the real estatebubble, which reduces the income when old. It implies a negative effect on bequest distributed tomiddle-aged households, who invest less in physical capital. This result questions the view arguingthat a government should increase taxation of unproductive assets, like real estate taxation, in orderto boost productive investment and, thus aggregate output.
The paper proceeds as follows. In the next section, we discuss the related literature. Section3 presents the model. Section 4 characterizes the co-existence of steady states without and with areal estate bubble and shows that the real estate bubble may enhance capital. Section 5 is devotedto the presentation of the model without altruism. Section 5 discusses the role of fiscal policy. Alast section provides concluding remarks, whereas technical details are relegated to an Appendix.
2 Related literature
Our paper is related to a recent growing literature on rational housing bubbles in overlappinggenerations (OLG, henceforth) models. The existence of such bubbles has been studied in OLGmodels first by Arce and Lopez-Salido (2011) and Basco (2014). They investigate the interplaybetween real estate bubbles and financial frictions, but without physical capital accumulation in
to entrepreneur’s income.
3
contrast to our paper. In their models, households deriving utility from housing services, butthey model the housing bubble as a shortage of assets in the economy, and not as a componentof the housing price as in our model. By the way, Arce and Lopez-Salido (2011) show that thehousing price is less than its fundamental value, i.e. the sum of discounted dividends, in the bubblyequilibrium. Zhao (2015) builds an OLG model with homeowners, who derive utility from housingservices, and investors, who do not. He shows that a housing bubble only exists in an equilibriumin which investors hold houses for resale purpose only, and not for dividend purpose either in termsof rents or in terms of utility. In a recent contribution, Huber (2018) shows that an housing bubble,as a component of the housing price, can exist in an OLG economy with housing services, butwithout capital accumulation. Her results does not rely on the existence of any financial frictions,but on the features of the housing market. At each period, the young generation is endowed witha part of the housing stock belonging to the old generation. Through this feature, she introducesa new bubble at each period. In contrast to Zhao (2015) and Huber (2018), we abstract ourselvesfrom modelling housing services through utility. This allows us to deeply focus on the the role ofaltruism in the existence of rational real estate bubble and on the interplay between rational realestate bubble and capital accumulation.
To the best of our knowledge, only Bosi et al. (2018) investigate the influence of altruism on theexistence of a bubble. They consider a two-period OLG model with altruism and an asset bringingnon-stationary positive dividends. They study the dynamics of capital stock and asset value. Incontrast to our paper, they consider a form of altruism rationalized by a bequest constraint andare not concerned with the design of fiscal policy.
This paper also belongs to the rational bubble literature studying the macroeconomic effects ofa bubble in OLG models with financial frictions. Farhi and Tirole (2012) and Martin and Ventura(2012) show that a bubble can have a crowd-in effect by promoting capital accumulation, and thusincreasing aggregate output through the existence of financial frictions, which generates a wealtheffect coming from the bubble. Even if we consider financial frictions, we differ from Martin andVentura (2012) because our analysis does not require bubble shocks, and from Farhi and Tirole(2012) because our mechanism is not based on the existence of a wealth effect. It relies on theexistence of altruistic households and, thus on inheritance. By receiving bequests, middle-agedhouseholds have a sufficiently high income to sustain a higher level of capital during bubbly episodes.Raurich and Seegmuller (2015) investigate the interplay between long-term productive investmentsin physical capital and short-term speculative ones in the asset bubble. Their mechanism is alsodifferent than ours, since it relies on selling short the bubble to invest more in capital. There isalso a large literature studying the existence of productive bubbles in infinite horizon models whichdiffers from our approach. For instance, Kocherlakota (2009), Hirano and Yanagawa (2017) andMiao and Wang (2018) build models with credit market imperfections. In contrast, Kamihigashi(2008) shows that bubbles can affect output positively in an economy with increasing social returnsand capital spirit.
Finally, only few other contributions analyse the effects of taxation on both the existence ofbubbles and on capital accumulation. Miao et al. (2015) are interested in the existence of a realestate bubble in an infinite-horizon model with production and credit constraints faced by theentrepreneurs. They show that a bubble can exist and reduce welfare. Some fiscal policies, such asproperty or transaction taxation, are able to eliminate the bubble. We differ from this paper sincewe consider a model with altruistic households facing credit constraints. Furthermore, we addressa different political issue. Raurich and Seegmuller (2015, 2017) also deal with fiscal policy issue, byanalysing the consequences of capital and labour income taxes on the existence and macroeconomic
4
effect of a bubble. In our paper, we consider an alternative source of taxation, namely a real estatetaxation.
3 Model
We consider an overlapping generations model with discrete time (t = 0, 1, ...,+∞). This economyconsists of two types of agents, households and firms.
3.1 Households
There is no population growth and, at each date t, a generation of unit size is born. Each generationlives for three periods.
Utility. Preferences are represented by an additively separable life-cycle utility function:
lnc1,t + βlnc2,t+1 + β2lnc3,t+2 + δεln(φwt+2 + xt+2) (1)
where c1,t, c2t+1, and c3,t+2 denote the consumption at time t, t+ 1, t+ 2 of a household born att, β ∈ (0, 1) the discount factor and δ > 0 the degree of altruism of a household. The parameterε ∈ {0, 1} allows us to consider the model with or without altruism.
Through these preferences, we introduce family altruism, defined as the situation in whichgeneration t cares about the next generation income. More precisely, old parents take care abouttheir middle-aged children: φwt+2 represents the labour income of a middle-aged agent born attime t+ 1 and xt+2 her bequest received from their old parents born at time t.
Budget constraints. In her first period of life, the household is young and supplies one unit oflabour inelastically remunerated at the wage wt. She can borrow an amount dyt on a credit market.With her wage and borrowing, she can consume an amount c1,t of final good and buy an asset ht+1,which is assumed to be intrinsically useless, at the price pt in units of consumption. We interpretthis asset as real estate. A positive price pt > 0 depicts a bubble on real estate.
At the second period of life, the household is middle-aged. Having a better productivity, shesupplies inelastically φ efficient units of labour remunerated at the wage wt+1. A middle-agedhousehold may receive a bequest xt+1 from their old parents. Bequests and labour income are usedto repay their debt at the gross rate Rdt+1, to consume c2,t+1 of final good, and to save through adiversified portfolio of physical capital kt+2 and deposits dmt+1.
At the third period of life, the household is old and retired. She uses the sales of their real estateasset bought at time t at price pt+2 and her remunerated savings from physical capital at the grossrate Rt+2 and from her deposit at the gross rate Rdt+2 to consume c3,t+2 units of final good andgive a positive bequest xt+2 to her children.4
c1,t + ptht+1 = wt + dyt (2)
c2,t+1 +Rdt+1dyt + dmt+1 + kt+2 = φwt+1 + εxt+1 (3)
c3,t+2 + εxt+2 = pt+2ht+1 +Rt+2kt+2 +Rdt+2dmt+1 (4)
xt+2 > 0 (5)
4We assume a full capital depreciation within a period.
5
Through these budget constraints, we assume that a household chooses first to invest in a longterm asset, real estate, that she resells when hold, and has access to the productive capital only atmiddle-age. Interpreting kt+2 as shares of firms, it symbolizes the idea that households first investin real estate, kept during all the life-cycle and used to transfer bequests to children. The loans tofinance this investment are reimbursed in middle-age. Then, agents are concerned with financialinvestments, through deposits and productive capital.
Borrowing constraint. Doing their investment in real estate, young households face a borrowingconstraint:
Rdt+1dyt ≤ θ1pt+1ht+1 + θ2φwt+1 (6)
with 0 ≤ θ1 < 1 and 0 ≤ θ2 < 1
Borrowing can only take place up to the point where the principal plus interest, i.e. Rdt+1dyt , is
secured by a fraction θ1 of the market value of real estate owned by the household at t+ 1, and afraction θ2 of her wage received at the period of debt repayment. If households repudiate their loanobligations, the lenders can seize a fraction θ1 of their real estate asset. This collateral constraintis consistent with standard lending criteria used in the mortgage market. This type of constraintgenerates two wealth effects one which is direct coming from future wage, and the other one whichis indirect by modifying the relative price of the asset used as collateral. Through this constraint,we make a distinction between a “fundamental” collateral (future wages) and a “bubbly” collateral(value of bubble asset).
Optimal behaviour. Let us assume ε = 1, i.e. agents are altruistic. A household derives heroptimal consumption choice (c1,t, c2,t+1, c3,t+2), her optimal portfolio choice (kt+2, dyt , dmt+1, ht+1),and her optimal bequest (xt+2) by maximizing her utility function (1) under her budget, bequestand borrowing constraints (2)-(6). We restrict our attention to an economy in which the borrowingconstraint is binding. The optimal behaviour of a household born at time t is summarized by thefollowing equations (see also Appendix 8.1):
β2
c3,t+2(pt+2 − θ1pt+1Rt+2) =
1
c1,t
(pt − θ1
pt+1
Rt+1
)(7)
1
c2,t+1= Rt+2
β
c3,t+2(8)
Rdt+2 = Rt+2 (9)
β2
c3,t+2=
δ
φwt+2 + xt+2(10)
θ1pt+1ht+1 + θ2φwt+1 = Rt+1dyt (11)
β2Rdt+1Rt+2c1,t < c3,t+2 (12)
xt+2 > 0 (13)
Because of the borrowing constraint, Eq. (7) depicts a modified intertemporal trade-off betweenthe first and third period consumptions. It requires that the following inequality must be satisfiedat each period t:
θ1pt+1
pt< Rt+1 <
1
θ1
pt+1
pt(14)
6
Since households face no borrowing constraints at their second and third period of life, Eq. (8)depicts the standard intertemporal trade-off between the second and third period consumptions. Intheir second period of life, households save through a diversified portfolio of deposits and physicalcapital. As the two assets are perfectly substitutes, they should provide the same return (Eq. (9)).Eq. (10) depicts the trade-off between the strictly positive bequests and third-period consumption.Since we restrict our attention to an economy with a binding borrowing constraint (Eq. (11)),inequality (12) should be satisfied.
3.2 Firms
The technology is characterized by the following aggregate production function:
Yt = Kαt L
1−αt ,with α ∈ (0, 1/2) (15)
where Lt is the total amount of efficient units of labour and Kt is the stock of physical capital inthe economy. Using a = K/L, the intensive production function is Yt/Lt = aαt , and competitivefactor prices are given by :
wt = (1− α)aαt ≡ w(at) (16)
Rt = αaα−1t ≡ R(at) (17)
Let s ≡ (1− α)/α(> 1). Using Eqs. (16) and (17), we obtain the ratio of factor prices:
wtRt
= sat (18)
3.3 Asset markets
The market of real estate asset is segmented in the sense that only young and old households cantrade this asset ht. As already explained, young households buy the asset in order to sell it whenthey will be old. Households behave as investors who buy real estate for resale purpose only, andnot for housing services.5 The supply of this asset is fixed and normalized to 2. Since this assethas no fundamental value, a bubble exists if pt > 0.
Deposits from middle-aged are used for loans of the young. This explains that loans and depositsprovide the same rate Rdt+1.
3.4 Intertemporal equilibrium
At date t = 0, the initial old household and initial middle-aged household hold one unit of realestate. Therefore, at each period, only one unit of financial asset is exchanged, implying thatht+1 = 1. Loans being equal to deposits, at each period of time, we have dyt = dmt ≡ dt.
5In an OLG model with households deriving utility from housing services (homeowners) and others doing not(investors), Zhao (2015) shows that a bubble in house price exists in an equilibrium if investors hold houses for resalepurpose only and not for dividend purpose either in terms of rents or in terms of utility. Based on this result, wedo not model the behaviour of an homeowner. This allows us to focus on the the role of altruism in the existence ofasset bubble. Arce and Lopez-Salido (2011) and Basco (2014) also study housing bubbles in overlapping generationsmodels. Nevertheless, they model housing bubble as a shortage of assets in the economy, and not as a component ofthe housing price.
7
Taking into account that the population size of a generation is normalized to one, labor measuredin efficient units is equal to Lt = 1 + φ. Then, equilibrium on the physical capital market meansthat capital per capita is given by kt = (1+φ)at. In addition, since physical capital and deposits areperfect substitutes for a household, they provide the same return at the equilibrium, i.e. Rt = Rdt .
Using Eq. (18), we deduce that Eq. (11) rewrites as:
dt =θ1pt+1
Rt+1+ θ2φsat+1 (19)
Then, using the above equilibrium conditions, the budget constraints (2)-(4), Eqs. (16) and(19), the trade-offs (7), (8) and (10) write:(θ1pt+1
Rt+1− pt
)(φsRt+2at+2 + xt+2) = δ (θ1pt+1Rt+2 − pt+2)
(sRtat + θ2φsat+1 + θ1
pt+1
Rt+1− pt
)(20)
β (φsRt+2at+2 + xt+2) = δRt+2 [φsRt+1at+1 (1− θ2)− (1 + φ+ φθ2s) at+2
+xt+1 − θ1(pt+1 +
pt+2
Rt+2
)](21)
β2
δ(φsRt+2at+2 + xt+2) = (1 + θ1) pt+2 + (1 + φ+ θ2φs)Rt+2at+2 − xt+2 (22)
From Eq. (22), we deduce:
xt+2 = δ(1 + θ1) pt+2 +
[1 + φ+
(θ2 − β2/δ
)φs]Rt+2at+2
β2 + δ(23)
Let us consider the following assumption:
Assumption 1 δ > δ, with δ ≡ β2 φs1+φ+φθ2s
.
Assumption 1 ensures a strictly non-negative bequests, i.e. xt+2 > 0, by imposing a sufficiently adegree of altruism δ. Substituting the expression of xt defined by Eq. (23) into Eqs. (20) and (21),the intertemporal trade-offs faced by households are summarized by the following equations:(θ1pt+1
Rt+1− pt
)(A1Rt+2at+2 +A2pt+2) = (θ1pt+1Rt+2 − pt+2)
(sRtat +A3at+1 + θ1
pt+1
Rt+1− pt
)(24)
B1pt+2 = Rt+2 (B2Rt+1at+1 −B3pt+1 −B4at+2) (25)
with
A1 ≡ [1 + φ+ (1 + θ2)φs] /(β2 + δ) > 0
A2 ≡ (1 + θ1)/(β2 + δ) > 0
A3 ≡ θ2φs > 0
B1 ≡ θ1 + (1 + θ1)β/(β2 + δ) > 0
B2 ≡[δ(1 + φ) + (δ − θ2β2)φs
]/(β2 + δ) > 0
B3 ≡ (β2θ1 − δ)/(β2 + δ)
B4 ≡ (1 + φ+ θ2φs) (β + β2 + δ)/(β2 + δ) + βφs/(β2 + δ) > 0
and Rt+1 given by Eq. (17).
8
Definition 1 Under Assumption 1, an intertemporal equilibrium with perfect foresight is a sequence(at, pt) ∈ R++ × R+, t = 0, 1, ...,+∞, such that the dynamic system (24)-(25) is satisfied, theborrowing constraint is binding and θ1
pt+1
pt< Rt+1 <
1θ1
pt+1
pt, where Rt+1 is defined by Eq. (17)
and a0 > 0 is given.
In the following, we will use the definition of the equilibrium to analyse the existence of a steadystate with a productive real estate bubble, i.e. a bubble associated to a larger level of capital andoutput per worker.6
4 Is the real estate bubble capital enhancing?
From Eq. (24), there exist two types of steady states: a bubbleless steady state (i.e. p = 0) and abubbly steady state (i.e. p > 0). We first characterize the bubbeless steady state, and second thebubbly. Then, we will be able to compare them to show that the bubble may enhance capital.
4.1 Bubbleless steady state p = 0
A bubbleless steady state is characterized by p = 0. Using (25), the gross interest rate R and thecapital per labour a are respectively given by:
R =B4
B2and a =
(αB2
B4
)1/(1−α)
(26)
Note that Assumption 1 ensures a non-negative gross interest rate, i.e. R > 0.
Proposition 1 Let δ ≡ β[(1 + φ + φθ2s)(1 + β) + φs(1 + θ2β)]/[φs(1 − θ2)]. Under Assumption1, there exist φ and θ2 > 0 such that there is a unique bubbleless steady state (p,R) with a binding
borrowing constraint characterized by p = 0 and R ∈ (0, 1) if δ is sufficiently high such that δ > δ,
φ > φ and θ2 < θ2.
Proof. See Appendix 8.2.
To understand that R < 1 is obtained, let us recall that only middle-aged households can investin physical capital. If households are sufficiently altruistic, then they will leave a high amount ofbequest to their middle-aged children, ceteris paribus. As savings are normal good, middle-agedhouseholds can save a significant amount in physical capital for a given wage w. This tends tolower the interest rate. If household cannot pledge a huge part of their income (weak θ2), the loanrepayment will be weak. For a given wage w, their income after loan repayment will be high, thusmiddle-aged household will increase their savings in physical capital, ceteris paribus. This tends tolower the interest rate. We finally note that a high enough φ means that labour income is moreredistributed to middle-aged. This facilitates a positive borrowing for the young agents, throughthe redistribution of income from the middle-age to the young.
6Since the economy is described by a four-dimensional dynamic system, we restrict our attention to the analysisof steady states and do not study the dynamics.
9
4.2 Bubbly steady state p > 0
A bubbly steady state (p,R, a) must satisfy the following system: G(R) = H(R)
p = RG(R)a(27)
with G(R) ≡ B2R−R
B1 +RB3(28)
and H(R) ≡ −(R− θ1)A1 + (1− θ1R)(sR+A3)
(R− θ1)(A2 + 1− θ1R)(29)
where R ≡ R(a) is given by Eq. (17).
The borrowing constraint is binding if Inequality (12) is satisfied. Combining with Eq. (7), abinding borrowing constraint implies R < 1 at the steady state. Moreover, Inequality (14) must besatisfied at the steady state. Therefore, at the bubbly steady state, R must belong to (θ1, 1).
According to the sign of B3, two types of bubbly steady states can exist: a steady state withunproductive real estate bubble (R < R or a < a) and a steady state with productive real estatebubble (R > R or a < a). Note that the steady state with productive real estate bubble couldoccur only if B3 < 0 and R > −B1/B3 > θ1.7 As a steady state with productive real estate bubbleis in accordance with the data, we will restrict our attention to the case for which R > −B1/B3.To ensure B3 < 0, we make the following assumption:
Assumption 2 δ > δ, with δ ≡ β2θ1.
Taking into account that there is a bubbleless steady state, we show the existence of a bubblysteady state with a higher level of production:
Proposition 2 Let θ1 ≡ (1 + φ + φθ2s)/(1 + φ + φs). Under Assumptions 1 and 2, there existsδ > 0 such that a unique steady state with productive real estate bubble (a, p) characterized byR ∈ (−B1/B3, R) and p > 0 coexists with the bubbleless steady state if φ is close to but larger than
φ, θ2 < θ2, θ1 < θ1 and δ sufficiently high such that δ > max{δ, δ, δ}. This means that a > a.
Proof. See Appendix 8.3.
We show that the real estate bubble is productive and the bubbly steady state coexists with thebubbleless one if R < 1. A low interest rate environment means that there is an excess of savings inthe economy without bubble, which provides a room for alternative stores of value. At the bubblysteady state, the real estate bubble provides a higher return equal to one. Therefore, this returncreates an incentive for households to hold the bubble, since other assets have lower returns andborrowing has a lower cost.
We also observe that, according to Proposition 2, a productive real estate bubble occurs ifhouseholds are sufficiently altruistic (i.e. high δ), the effects of fundamental and bubbly collateralsare positive but not too important (i.e. θ2 < θ2 and θ1 < θ1), and an important share of laborincome is distributed to middle-aged agents (i.e. φ high enough).
7According to (27), a stationary bubble means G(R) > 0.
10
To understand the main mechanisms explaining that a real estate bubble enhances capital, werewrite the first and second period budget constraints combined with the borrowing constraint atthe bubbly steady state:
c1 + p (1− θ1/R) = w(1 + φθ2/R) (30)
c2 +1 +R
R(θ1p+ θ2φw) + (1 + φ)a = φw + x (31)
A sufficiently altruistic household want to leave high bequests to their middle-aged children.Investing in the real estate bubble when young and reselling it when old allows her to increasebequests, which pushes up the income of their children at middle-age. This additional income isin particular used to raise the investment in capital, which explains that the real estate bubble isproductive.
We deduce from Proposition 2 that a real estate bubble is productive under quite similar condi-tions on θ1 and θ2. This suggests that fundamental and bubbly collaterals play similar roles on theexistence of the real estate bubble, even if the interpretations can be slightly different. By inspectionof Eq. (30), higher θ1 and θ2 increase the role of collateral and allow the household to borrow morewhen young. Indeed, taking into account the credit constraint, a higher θ2 generates a positiveincome effect through the wage, whereas a higher θ1 implies a lower price of the real estate bubble.Both these effects will promote a higher price of the real estate asset, which increases wealth whenold, bequests and therefore investment in capital of the middle-aged households. However, usingEq. (31), we observe that at middle-age, a higher θ1 or θ2 means a higher debt reimbursed anda larger funding of the borrowing of young agents. Both these effects have a negative impact oncapital accumulation and explain that our result does not hold when either θ1 or θ2 is too close toone.
We finally note that a high enough value of φ contributes to ensure a significant income atmiddle-age to finance investments and the reimbursement of the loans contracted while being young.
Of course, the most important ingredient for our explanation of productive real estate bubble lieson altruism. The sufficient conditions we get are based on a sufficiently high degree of altruism. Toclarify the crucial role played by altruism to generate a real estate bubble which enhances capital,we analyse now the model without altruism and show that the bubble crowds out capital insteadof raising it.
5 The crucial role of altruism
The model without altruism corresponds to the case where ε = 0. Its analysis is quite similar tothe model with altruism. Therefore, for the sake of brevity, we present the main derivations only.
A household maximizes the utility (1) under the budget constraints (2)-(4) and the borrowingconstraint (6). Focusing on solutions with a binding borrowing constraint, we get the following
11
optimal conditions:
β2
c3,t+2(pt+2 − θ1pt+1Rt+2) =
1
c1,t
(pt − θ1
pt+1
Rt+1
)(32)
1
c2,t+1= Rt+2
β
c3,t+2(33)
Rdt+2 = Rt+2 (34)
θ1pt+1ht+1 + θ2φwt+1 = Rt+1dyt (35)
β2Rdt+1Rt+2c1,t < c3,t+2 (36)
We still need to have θ1pt+1
pt< Rt+1 <
1θ1
pt+1
pt. An equilibrium satisfies ht = 1, kt = (1 + φ)at,
dyt = dmt = dt. Using Eqs. (2)-(4), (6) and the trade-offs (32) and (33), we get:
pt+2(1 + θ1) +Rt+2at+2(1 + φ+ φθ2s) = βRt+2 [φsRt+1at+1 (1− θ2)
− (1 + φ+ φθ2s) at+2 − θ1(pt+1 +
pt+2
Rt+2
)](37)(
pt − θ1pt+1
Rt+1
)[pt+2(1 + θ1) +Rt+2at+2(1 + φ+ φθ2s)] = β2 (pt+2 − θ1pt+1Rt+2)(
sRtat + θ2φsat+1 + θ1pt+1
Rt+1− pt
)(38)
and the condition for a binding borrowing constraint rewrites:
pt+2(1 + θ1) +Rt+2at+2(1 + φ+ φθ2s) > β2Rt+1Rt+2
(sRtat + θ2φsat+1 + θ1
pt+1
Rt+1− pt
)(39)
A bubbleless steady state with selfish households (ps, Rs) is characterized by p
s= 0. Using
(37), the gross interest rate Rs is given by:
Rs =1 + β
β
1 + φ+ θ2φs
(1− θ2)φs(40)
Moreover, using (39), the borrowing constraint is binding if and only if Fs(φ) > 0, with:
Fs(φ) ≡ (1− θ2)2sφ2 − (1 + β)[1 + β + φ(1 + β + θ2s(1 + 2β))] (41)
We can easily see that Fs(0) < 0 and Fs(+∞) = +∞. Therefore, there exists φs> 0 such that
Fs(φ) > 0 for all φ > φs.
Proposition 3 There exists φs> 0 such that there is a unique bubbleless steady state Rs with
selfish households with a binding borrowing constraint if φ > φs.
Let us focus now on a bubbly steady state with selfish households (ps, Rs, as). Using Eqs. (37)and (38), it satisfies:
psRsas
= G0(Rs) = H0(Rs) (42)
12
with G0(R) ≡ βφs(1− θ2)R−Rs
1 + θ1(1 + β) + θ1βR(43)
and H0(R) ≡ β2(1− θ1R)(sR+ θ2φs)− (R− θ1)(1 + φ+ φθ2s)
(R− θ1)(1 + θ1 + β2 − β2θ1R)(44)
We immediately see that ps > 0 means that G0(Rs) > 0, which implies that Rs > Rs.
Proposition 4 Every bubbly steady state Rs with selfish households with a binding borrowing con-straint is characterized by Rs > Rs. This means that without altruism, a stationary real estatebubble can never be productive.
This result confirms that altruism plays a key role for the occurrence of productive real estatebubbles. In the absence of altruism and bequest, middle-aged households have not a high enoughincome to sustain a higher level of capital when there is a real estate bubble.
6 Real estate taxation
There is a recurrent debate discussing whether it is better to tax real estate rather than capital toaccumulate more productive assets (see Leung, 2004). Using our model, we provide an argumentshowing that a higher tax on real estate diminishes capital. Therefore, it is not clear cut that oneshould recommend higher taxes on real estate.
We extend the model with altruism to take into account a tax on real estate. The main ideais to discuss whether a higher tax on real estate does or not generate a higher level of capital.Accordingly, we introduce a proportional tax on real estate τh(< 1), paid by the consumer whenold. Despite the fact that it distorts the relative prices, this tax is used to finance a public goodthat affects neither the preferences, nor the production.
With these new ingredients, the budget constraints (2)-(4) write:
c1,t + ptht+1 = wt + dyt (45)
c2,t+1 +Rdt+1dyt + dmt+1 + kt+2 = φwt+1 + xt+1 (46)
c3,t+2 + xt+2 = (1− τh)pt+2ht+1 +Rt+2kt+2 +Rdt+2dmt+1 (47)
while the utility is still given by (1) with ε = 1.There is a government that uses the tax revenues to finance government spendings Gt according
to the following balanced budget:Gt = τhptht−1 (48)
Since the tax rate τh is constant, public spendings Gt vary to balance the budget at each periodof time.
The rest of the model is similar to the one without taxation. Therefore, households maximizeutility (1) under the budget constraints (45)-(47) and the borrowing constraint (6). The optimalbehaviour of a household is solved in Appendix 8.4. By continuity with respect to the case withouttaxation, the following inequalities should be satisfied:
θ1pt+1
pt< Rt+1 <
1
θ1(1− τh)
pt+1
pt(49)
13
As in Section 2.5, an equilibrium satisfies ht = 1, kt = (1 + φ)at and dyt = dmt = dt. Using Eq.(47) and β2(φwt+2 + xt+2) = δc3,t+2, we deduce that:
xt+2 = δ(1− τh + θ1) pt+2 +
[1 + φ+
(θ2 − β2/δ
)φs]Rt+2at+2
β2 + δ(50)
Under Assumption 1, the bequest is positive whatever the tax rate and the bubble are. Followingthe same methodology than in Section 2.5, an equilibrium is defined by the following two equations:[
pt − θ1pt+1
Rt+1
](A1Rt+2at+2 + A2pt+2) = [(1− τh)pt+2 − θ1pt+1Rt+2](
sRtat +A3at+1 + θ1pt+1
Rt+1− pt
)(51)
pt+2B1 = Rt+2(B2Rt+1at+1 − B3pt+1 −B4at+2) (52)
with
A2 ≡1− τh + θ1β2 + δ
, B1 ≡ β1− τh + θ1β2 + δ
+ θ1, B3 ≡β2θ1 − δ(1− τh)
β2 + δ
and A1, A3, B2 and B4 defined as in Section 2.5.Of course, a bubbleless steady state is not affected by the real estate tax rate τh. In contrast,
a bubbly steady state with taxation (pT , RT , aT ) must satisfy the following system which dependson τh:
pTRTaT
= GT (RT ) = HT (RT ) (53)
with GT (R) ≡ B2R−RTB1 +RB3
(54)
and HT (R) ≡ (1− τh − θ1R)(sR+A3)− (R− θ1)A1
(R− θ1)(A2 + 1− τh − θ1R)(55)
The borrowing constraint is binding at the bubbly steady state with taxation if RT <√
1− τh.8
Moreover, Inequality (49) must be satisfied at the steady state, i.e. θ1 < RT < (1− τh)/θ1.At this stage, we consider that there exists a bubbly steady state with taxationRT ∈ (−B1/B3, RT )
whose existence and uniqueness are shown in a similar way than in Proposition 2.9 We investigatethe effect of taxes on such an equilibrium. The role of the tax on real estate τh is summarized inthe following proposition:
Proposition 5 Under Assumption 1, consider that there exists a unique bubbly steady state withtaxation RT and a binding borrowing constraint. Assuming δ > β/θ1, we show that RT is increasingin τh, meaning that aT is decreasing in τh.
Proof. See Appendix 8.5.
8Using Appendix 8.4, the borrowing constraint is binding combining Inequality (90) and Eq. (85).9Of course, by continuity, this is obvious if the tax rate is low enough.
14
This proposition shows that an increase of the tax on real estate has a negative effect oninvestment in productive capital. An increase of τh reduces the incentive to buy the real estatebubble, which reduces the income when old. Thus, it has a negative effect on bequest distributedto middle-aged agents, who invest less in capital.
This result provides an argument in favour of a reduction of real estate taxation, which may bequite significant, as in France (see Conseil des Prelevements Obligatoires, 2018).
7 Concluding remarks
In this paper, we present a three-period OLG model with capital accumulation, debt, real estateand bequests. Our model presents three features. First, households realise different investments atdifferent period of life. They borrow when young to buy a real estate asset for their old age andsave when middle-aged to transfer resources to their old age. Second, households face a bindingborrowing constraint. Because of credit market imperfections, they must pledge their wages andtheir real estate asset as collaterals. Third, households behave altruistically by leaving bequests totheir children.
We exhibit the existence of a steady state with a real estate bubble accompanied by a boomin physical capital and aggregate output. This result mainly relies on the existence of altruism inour model. Without altruism, middle-aged households have not a high enough income to sustain ahigher level of capital in the presence of a real estate bubble. There is no room for such a bubble.Moreover, we show that a productive real estate bubble exists whatever the type of collateral(fundamental or bubbly collateral) required by lenders on the credit market.
We further analyse the effects of real estate taxation on capital accumulation. We show thatan increase of real estate taxation dampens capital accumulation, which is harmful for the realeconomy. Therefore, our result provides an argument against the implementation of taxation onunproductive assets as real estate.
8 Appendix
8.1 Households’ behaviour
We maximize the Lagrangian function:
lnc1,t + βlnc2,t+1 + β2lnc3,t+2 + δln(φwt+2 + xt+2)
+λ1,t (wt + dyt − c1,t − ptht+1) + λ2,t+1
(φwt+1 + xt+1 − c2,t+1 −Rdt+1d
yt − kt+2 − dmt+1
)+λ3,t+2
(pt+2ht+1 +Rdt+2d
mt+1 +Rt+2kt+2 − c3,t+2 − xt+2
)+λ4,t+1
(θ1pt+1ht+1 + θ2φwt+1 −Rdt+1d
yt
)with respect to (c1,t, c2,t+1, c3,t+2, kt+2, d
yt , d
mt+1, ht+1, xt+2, λ1,t, λ2,t+1, λ3,t+2, λ4,t+1).
15
First-order conditions are given by:
c1,t :1
c1,t= λ1,t (56)
c2,t+1 :β
c2,t+1= λ2,t+1 (57)
c3,t+1 :β2
c3,t+2= λ3,t+2 (58)
ht+1 : ptλ1,t − pt+2λ3,t+2 − θ1pt+1λ4,t+1 = 0 (59)
xt+2 : λ3,t+2 −δ
φwt+2 + xt+2= 0 (60)
dyt : λ1,t = Rdt+1(λ2,t+1 + λ4,t+1) (61)
dmt+1 : λ2,t+1 = Rdt+2λ3,t+2 (62)
kt+2 : λ2,t+1 = Rt+2λ3,t+2 (63)
λ4,t+1
(θ1pt+1ht+1 + θ2φwt+1 −Rdt+1dt
)= 0 (64)
From Eqs. (62)-(63), we get : Rt+2 = Rdt+2. Physical capital and deposit are perfect substitutes.From Eqs. (61) and (63), one has:
λ4,t+1 =λ1,tRdt+1
−Rt+2λ3,t+2 (65)
We restrict our attention on the case where λ4,t+1 > 0. Substituting Eqs. (56)-(58) and (65) intoEqs. (59)-(63), one has :
β2
c3,t+2(pt+2 − θ1pt+1Rt+2) =
1
c1,t
(pt − θ1
pt+1
Rt+1
)(66)
β2
c3,t+2=
δ
φwt+2 + xt+2(67)
1
c2,t+1= Rdt+2
β
c3,t+2(68)
θ1pt+1ht+1 + θ2φwt+1 = Rdt+1dt (69)
8.2 Proof of Proposition 1
One can easily show that R < 1 if and only if δ > δ. The borrowing constraint is binding ifc3 > β2R2c1. This inequality is equivalent to:
A1 > R (sR+A3) (70)
Since R < 1 under δ > δ, inequality (70) is satisfied if
A1 > sR+A3 (71)
16
We can write A1 and sR+A3 as continuous functions of φ and θ2:
A1 =1 + φ [1 + (1 + θ2)s]
β2 + δ≡ LHS(φ, θ2) (72)
sR+A3 = sβ + β2 + δ + φ[(1 + θ2s)(β + β2 + δ) + βs]
δ + φ[(1 + s)δ − β2θ2s]+ φθ2s ≡ RHS(φ, θ2) (73)
Note that
LHS(φ, 0) =1 + φ [1 + s]
β2 + δ(74)
RHS(φ, 0) = sβ + β2 + δ + φ[β(1 + s) + β2 + δ]
δ + φ(1 + s)δ(75)
For θ2 = 0, Inequality (70) is satisfied if LHS(φ, 0) > RHS(φ, 0). Note that LHS(φ, 0) isan increasing function of φ and RHS(φ, 0) a decreasing function of φ. We have LHS(+∞, 0) >RHS(+∞, 0) and LHS(1, 0) < RHS(1, 0) if δ is sufficiently high. Therefore, it exists a thresholdφ0> 1 such that for φ > φ
0, LHS(φ, 0) > RHS(φ, 0). Note that LHS(φ, θ2) and RHS(φ, θ2)
are both increasing functions of θ2. Moreover, for δ high enough, LHS(+∞, 1) < RHS(+∞, 1).By continuity, there exist thresholds θ2 ∈ (0, 1) and φ > 1 such that for θ2 < θ2 and φ > φ,
LHS(φ, θ2) > RHS(φ, θ2). Therefore, if δ is sufficiently high such that δ > δ, φ > φ and θ2 < θ2,Proposition 1 follows.
8.3 Proof of Proposition 2
A bubbly steady state R is a solution of G(R) = H(R), where G(R) and H(R) satisfy (28)-(29).We start by determining the admissible range of values for R. We have already seen that
θ1 < R < 1. We remind that we restrict our attention to the existence of a productive bubble (i.e.R < R). As p = RG(R)a, B1 +RB3 must be negative to get a positive bubble.
As B1 = θ1 + (1 + θ1)β/(β2 + δ) > 0, B1 + RB3 < 0 implies R > −B1/B3 with B3 =(β2θ1 − δ)/(β2 + δ) < 0. Note that under Assumption 2 (δ > δ), B3 < 0 and −B1/B3 > θ1. Then,the possible admissible range of values for productive bubbles is R ∈ (−B1/B3, R), with R < 1.
To prove the existence of a stationary solution R ∈ (−B1/B3, R), we use the continuity of G(R)and H(R). Note that G(R) has a vertical asymptote at R = −B1/B3 and is a decreasing andcontinuous function of R when R increases to R. H(R) has two vertical asymptotes at R = θ1 <−B1/B3 and R = (1 +A2)/θ1 > 1. From Eqs. (28) and (29), we determine the boundary values ofG(R) and H(R):
limR→−B1/B3
G (R) = +∞
limR→R
G (R) = 0
limR→−B1/B3
H (R) =(B1/B3 + θ1)A1 + (1 + θ1B1/B3)(−sB1/B3 +A3)
(−B1/B3 − θ1)(A2 + 1 + θ1B1/B3)
limR→R
H (R) =−(R− θ1)A1 + (1− θ1R)(sR+A3)
(R− θ1)(A2 + 1− θ1R)
17
We have limR→−B1/B3H (R) < limR→−B1/B3
G (R). The existence of a steady state R ∈(−B1/B3, R), solving G(R) = H(R), is ensured by 0 < H(R), namely:
A1 <1− θ1RR− θ1
(sR+A3) (76)
Note that (1− θ1R)/(R− θ1) > 1 if and only if R < 1. Using the proof of Proposition 1, Inequality(76) is in accordance with Inequality (71) and satisfied by continuity argument if φ close but largerthan φ and θ2 ∈ (0, θ2), because it ensures that A1 is larger but close to sR+A3.
We ensure now that the set (−B1/B3, R) is non-empty. The set (−B1/B3, R) is non-empty ifand only if −B1/B3 < R with:
−B1
B3=
δθ1 + β + θ1β(1 + β)
δ − β2θ1≡ LHS(δ, θ1) (77)
R =(1 + φ+ φθ2s)(δ + β + β2) + βφs
δ(1 + φ+ φs)− θ2β2φs≡ RHS(δ) (78)
Note that LHS(δ, θ1) and RHS(δ) are decreasing functions of δ and we have that:
limδ→+∞
LHS (δ, θ1) = θ1
limδ→+∞
RHS (δ) =1 + φ+ φθ2s
1 + φ+ φs< 1
We deduce that limδ→+∞ LHS (δ, θ1) < limδ→+∞RHS (δ) if and only if:
θ1 <1 + φ+ φθ2s
1 + φ+ φs≡ θ1
If θ1 < θ1, there exists a threshold δ > 0 such that for δ > δ, LHS(δ, θ1) < RHS(δ). Therefore, ifθ1 < θ1 and δ > δ, the set (−B1/B3, R) is nonempty.
To conclude, if φ is close but larger than φ, θ2 < θ2, θ1 < θ1 and δ sufficiently high such that
δ > max{δ, δ, δ}, there exists a steady state R ∈ (−B1/B3, R) which coexists with the bubblelessone R.
To prove uniqueness, we observe that the equation H(R) = G(R) is equivalent to a poly-nomial of degree 3, i.e. has at most three solutions. Since limR→(1+A2)/θ1 H (R) = −∞ andlimR→(1+A2)/θ1 G (R) is finite, we have limR→(1+A2)/θ1 H (R) < limR→(1+A2)/θ1 G (R). We haveestablished that limR→RH (R) > limR→RG (R). This means that an odd number of solutions tothe equation H(R) = G(R) belong to (R, (1 + A2)/θ1). Since we also have limR→−B1/B3
G (R) >limR→−B1/B3
H (R), there are an odd number of solutions toH(R) = G(R) that belong (−B1/B3, R).
Since there are at most three solutions, the solution R in this last interval is unique.
8.4 Households’ behaviour in the model with taxation
We maximize the Lagrangian function:
lnc1,t + βlnc2,t+1 + β2lnc3,t+2 + δln[φwt+2 + xt+2]
+λ1,t (wt + dyt − c1,t − ptht+1) + λ2,t+1
[φwt+1 + xt+1 − c2,t+1 −Rdt+1d
yt − kt+2 − dmt+1
]+λ3,t+2
[(1− τh)pt+2ht+1 +Rdt+2d
mt+1 +Rt+2kt+2 − c3,t+2 − xt+2
]+λ4,t+1
(θ1pt+1ht+1 + θ2φwt+1 −Rdt+1d
yt
)18
with respect to (c1,t, c2,t+1, c3,t+2, kt+2, dyt , d
mt+1, ht+1, xt+2, λ1,t, λ2,t+1, λ3,t+2, λ4,t+1). First-order
conditions are given by:
1
c1,t= λ1,t,
β
c2,t+1= λ2,t+1,
β2
c3,t+2= λ3,t+2 (79)
ptλ1,t = (1− τh)pt+2λ3,t+2 + θ1pt+1λ4,t+1 (80)
λ3,t+2 =δ
φwt+2 + xt+2(81)
λ1,t = Rdt+1(λ2,t+1 + λ4,t+1) (82)
λ2,t+1 = Rdt+2λ3,t+2, λ2,t+1 = Rt+2λ3,t+2 (83)
From Eq. (83), we get Rt+2 = Rdt+2. From Eqs. (82) and (83), one has:
λ4,t+1 =λ1,tRdt+1
−Rdt+2λ3,t+2 (84)
We restrict our attention on the case where λ4,t+1 > 0, i.e. λ1,t > Rdt+1Rdt+2λ3,t+2. Substituting
Eq. (79) into Eqs. (80), (81) and (83), we deduce:
β2
c3,t+2[(1− τh)pt+2 − θ1pt+1Rt+2] =
1
c1,t
[pt − θ1
pt+1
Rt+1
](85)
1
c2,t+1= Rt+2
β
c3,t+2(86)
Rdt+2 = Rt+2 (87)
β2
c3,t+2=
δ
φwt+2 + xt+2(88)
θ1pt+1ht+1 + θ2φwt+1 = Rt+1dyt (89)
β2Rt+1Rt+2c1,t < c3,t+2 (90)
xt+2 > 0 (91)
8.5 Proof of Proposition 5
The existence of the steady state is obtained because GT (R) > HT (R) when R tends to −B1/B3
and GT (R) < HT (R) when R tends to RT . Uniqueness allows to deduce that at the steady state,we have G′T (R) < H ′T (R).
Using Eq. (53), we have:
dRTd(1− τh)
=
∂GT (RT )∂(1−τh) −
∂HT (RT )∂(1−τh)
H ′T (RT )−G′T (RT )(92)
We start by computing ∂GT (RT )/∂(1− τh). Note that B2 and B4 do not depend on τh, whilewe have:
∂(B1 +RB3)
∂(1− τh)=β − δRβ2 + δ
< 0
19
because R > θ1 and δ > β/θ1. We deduce that:
∂GT (RT )
∂(1− τh)= − GT (RT )
B1 +RT B3
β − δRTβ2 + δ
< 0 (93)
because B1 +RT B3 < 0.Using Eq.(55), HT (R) can be rewritten as follows:
HT (R) =β2 + δ
R− θ1(1− τh − θ1R)(sR+A3)− (R− θ1)A1
(1− τh)(1 + β2 + δ) + θ1[1− (β2 + δ)R](94)
Obviously, ∂HT (RT )/∂(1− τh) has the sign of its numerator, i.e.
(sRT +A3)θ1(1 +RT ) + (1 + β2 + δ)A1
(RT − θ1
)> 0
Using Eq. (92), we deduce that dRT /d(1 − τh) < 0. This means that RT increases with τh,whereas aT decreases with τh.
References
[1] Arce, O. and Lopez-Salido, D. (2011), “Housing bubbles”, American Economic Journal:Macroeconomics 3, 212-241.
[2] Basco, S. (2014), “Globalization and financial development: A model of the dot-com and thehousing bubbles”, Journal of International Economics 92, 78-94.
[3] Bosi, S., Ha-Huy, T., Le Van C., Pham C.-T. and N.-S. Pham (2018), “Financial bubblesand capital accumulation in altruistic economies”, Journal of Mathematical Economics 75,125-139.
[4] Campbell, J. and G. Mankiw (1989), “Consumption, income and interest rates: Reinterpret-ing the time series evidence,”In: O. Blanchard and S. Fisher (Eds.) NBER MacroeconomicsAnnual, MIT Press, Cambridge MA, 185-216.
[5] Conseil des Prelevements Obligatoires (2018), “Les prelevements obligatoires sur le capital desmenages”, 1-140.
[6] Crook, J. and S. Hochguertel (2005), “Household debt and credit constraints: Evidence fromOECD countries”, Credit Research Center, University of Edinburgh Working Paper Series No.05/02.
[7] Dynan, K. E., J. Skinner and S. P. Zeldes (2004), “Do the rich save more?”, Journal of PoliticalEconomy 112, 397-444.
[8] Fahri, E. and J. Tirole (2012), “Bubbly liquidity”, Review of Economic Studies 79, 678-706.
[9] Hirano, T. and N. Yanagawa (2017), “Asset bubbles, endogenous growth, and financial fric-tions”, Review of Economic Studies 84, 406-443.
[10] Huber, S. (2018), “Preference for housing services and rational house price bubbles”, mimeo.
20
[11] Iacoviello, M. (2004), “Consumption, house prices, and collateral constraints: A structuraleconometric analysis,”Journal of Housing Economics 13, 304-320.
[12] IMF - International Monetary Fund (2015), World Economic Outlook, Uneven Growth: Short-and Long-Term Factors.
[13] Jappelli, T. (1990), “Who is credit constrained in the US economy?,”Quarterly Journal ofEconomics 105, 219-234.
[14] Kamihigashi, T. (2008): “The spirit of capitalism, stock market bubbles and output fluctua-tions ”, International Journal of Economic Theory 4, 3-28.
[15] Kiyotaki, N. and J. Moore (1997), “Credit cycles ”, Journal of Political Economy 4, 211-248.
[16] Kocherlakota, N.R. (2009), “Bursting bubbles: Consequences and cures”, University of Min-nesota working paper.
[17] Kotlikoff L. J. and L. H. Summers (1981), “The role of intergenerational transfers in aggregatecapital accumulation”, Journal of Political Economy 89, 706-732.
[18] Leung, C. (2004), “Macroeconomics and housing: A review of the literature ”, Journal ofHousing Economics 13, 249-267.
[19] Loi n0 2017-1837 du 30 decembre 2017 de finances pour 2018.
[20] Ludvigson, S. (1998), “Consumption and credit: A model of time-varying liquidity constraints”,The Review of Economics and Statistics 81, 434-447.
[21] Lustig, H. and S. Van Nieuwerburgh (2010), “How much does household collateral constrainregional risk sharing?”, Review of Economic Dynamics 13, 265-294.
[22] Marcet, A. and K. Singleton (1999), “Equilibrium asset prices and saving of heterogeneousagents in the presence of incomplete markets and portfolio constraints”, Macroeconomic Dy-namics 3, 243-277.
[23] Martin, A. and J. Ventura (2011), “Theoretical notes on bubbles and the current crisis”, IMFEconomic Review 59, 6-40.
[24] Martin, A. and J. Ventura (2012), “Economic growth with bubbles”, American EconomicReview 102, 3033-58.
[25] Martin, A. and J. Ventura (2016), “Managing credit bubbles ”, Journal of the European Eco-nomic Association 14, 753-789.
[26] Mera, K. and B. Renaud (2000), Asia’s Financial Crisis and the Role of Real Estate, Taylorand Francis, New-York, USA.
[27] Miao, J. and P. Wang (2018), “Asset bubbles and credit constraints”, forthcoming in AmericanEconomic Review.
[28] Miao, J., Wang. P and J. Zhou (2015), “Housing bubbles and policy analysis”, 2015 MeetingPapers 1056, Society for Economic Dynamics.
21
[29] Raurich, X. and Seegmuller (2015), “On the interplay between speculative bubbles and pro-ductive investment”, AMSE Working Paper 2015-42.
[30] Raurich, X. and Seegmuller (2017), “Income distribution by age group and productive bubbles”,AMSE Working Paper 2017-40.
[31] Tirole, J. (1985), “Asset bubbles and overlapping generations”, Econometrica 53, 1071-1100.
[32] Zhao, B. (2015), “Rational housing bubble”, Economic Theory 60, 141-201.
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